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Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions. (English) Zbl 0762.49017
The paper is concerned with the optimal design problem $\inf_ A \int_ \Omega j\bigl(x,u_ A(x)\bigr) dx,$ where $$\Omega{}$$ is a given domain of $${\mathbf R}^ n$$ ($$n\geq{} 2$$), $$A$$ varies in the open subsets of $$\Omega{}$$ and $$u_ A\in{} H^ 1_ 0\bigl(A\bigr)$$ is the solution of the problem $\Delta{} u_ A=-f\quad\quad\hbox{\mathrm in }A.$ The cost functions $$j$$ and $$f\in{} L^ 2\bigl(\Omega{}\bigr)$$ are given. In general, the infimum is not attained. By using the approximation results of G. Dal Maso and U. Mosco [Appl. Math. Optim. 15, 15-63 (1987; Zbl 0644.35033)], the authors are able to find a relaxed formulation of the problem which always admits a solution. Moreover, they find a set of necessary conditions for optimality. Some of these were already known in the literature [see for instance O. Pironneau, “Optimal shape design for elliptic systems” (1984; Zbl 0534.49001)].
Reviewer: L.Ambrosio (Roma)

##### MSC:
 49Q10 Optimization of shapes other than minimal surfaces 49K15 Optimality conditions for problems involving ordinary differential equations 35D05 Existence of generalized solutions of PDE (MSC2000)
##### Citations:
Zbl 0644.35033; Zbl 0534.49001
Full Text:
##### References:
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